A class of higher‑rank Chern‑Simons theories, defined in 2+1 dimensions, has been shown to support exotic edge dynamics. When a planar boundary is introduced, the theory yields two types of conserved currents on the edge: a traceless rank‑two current obeying a fractonic double‑divergence law, and an ordinary vector current with a standard U(1) Kac‑Moody algebra.

The authors extract these boundary currents by taking the zeroth and first normal moments of a scalar Ward identity that follows from the theory’s gauge symmetry. Differentiating the resulting identities with respect to external sources produces the two integrated boundary equations. The zeroth moment gives the traceless rank‑two tensor current, while the first moment brings the normal‑tangential field into play and yields the vector current.

A key consistency check involves the bulk‑boundary matching of energy signs. The analysis shows that positive‑energy matching can only be achieved when an auxiliary field, denoted ϕ, is set to zero. If ϕ≠0, the compatibility conditions clash with the Hamiltonian positivity requirement, preventing a coherent bulk‑boundary description. In the ϕ=0 sector, the boundary equations align with the bulk equations on a chiral subsector, satisfying both equations of motion and ensuring a well‑behaved physical theory.

These results provide a theoretical framework for exploring fracton physics on surfaces and may guide future investigations into higher‑rank gauge theories with physical applications.